Rewriting and solving a real-world exponential model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation
10x=500 represents this situation, where
x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

The amount of energy released from one earthquake was
8,500 times greater than the amount of energy released from another. The equation
10x=8500 represents this situation, where
x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Using natural logarithms

The most frequently used base for logarithms is
e. Base
e logarithms are important in calculus and some scientific applications; they are called
natural logarithms . The base
e logarithm,
loge(x), has its own notation,
ln(x).

Most values of
ln(x) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,
ln1=0. For other natural logarithms, we can use the
ln key that can be found on most scientific calculators. We can also find the natural logarithm of any power of
e using the inverse property of logarithms.

A General Note

Definition of the natural logarithm

A
natural logarithm is a logarithm with base
e. We write
loge(x) simply as
ln(x). The natural logarithm of a positive number
x satisfies the following definition.